On the homological mirror symmetry conjecture for pairs of pants

More by Nick Sheridan

Abstract

The $n$-dimensional pair of pants is defined to be the complement
of $n + 2$ generic hyperplanes in $\mathbb{CP}n$. We construct an immersed
Lagrangian sphere in the pair of pants and compute its
endomorphism $A_\infty$ algebra in the Fukaya category. On the level
of cohomology, it is an exterior algebra with $n+2$ generators. It is
not formal, and we compute certain higher products in order to determine
it up to quasi-isomorphism. This allows us to give some
evidence for the Homological Mirror Symmetry conjecture: the
pair of pants is conjectured to be mirror to the Landau-Ginzburg
model $(\mathbb{C}^{n+2},W)$, where $W = z_1...z_{n+2}$. We show that the endomorphism
$A_\infty$ algebra of our Lagrangian is quasi-isomorphic to
the endomorphism dg algebra of the structure sheaf of the origin
in the mirror. This implies similar results for finite covers of the
pair of pants, in particular for certain affine Fermat hypersurfaces.